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The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of (ψ, β )-differentiable functions in the integral metric. The Poisson integral is taken as a positive linear operator, being the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disc. The Poisson integral refers to operators with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: optimization methods and calculus of variations, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of (ψ, β )-differentiable functions are generalizations of the well-known Sobolev, Weyl–Nagy, etc. classes in optimization problems, on which the asymptotic properties of solutions of Laplace equations in the unit disc are analyzed. The problem solved in the paper will make it possible to generate high-quality mathematical models of many natural and social processes. [ABSTRACT FROM AUTHOR]

In this paper we consider a iterative system of two-point boundary value problems with integral boundary conditions having n singularities and involve an increasing homeomorphism, positive homomorphism operator. By applying Hölder's inequality and Krasnoselskii's cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of denumerably many positive solutions. Finally we provide an example to check validity of our obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study partial integral operators in Kaplansky-Hilbert module. A modular and cyclic modular spectrum of such operators is described, and a corresponding example is given. In addition, a variant of the Mercer theorem for partial integral operators with a symmetric and positive kernel has been proved. [ABSTRACT FROM AUTHOR]

We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy. [ABSTRACT FROM AUTHOR]

A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C(S^1)^{(n)} of n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}), where \mathcal {H} is an arbitrary Hilbert space and C(S^1)_{(n)} is the operator system dual of C(S^1)^{(n)}. Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from \mathcal {B}(\mathcal {H}) when \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps \psi on \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that \psi ^{(n)}(x) is positive whenever x is a positive n\times n Toeplitz matrix with entries from \mathcal {B}(\mathcal {H}). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality. [ABSTRACT FROM AUTHOR]

Let 픸 = ( A O O A ) be a 2 × 2 diagonal operator matrix whose each diagonal entry is a bounded positive (semi-definite) linear operator A acting on a complex Hilbert space ℋ . In this paper, we derive several 픸 -numerical radius inequalities for 2 × 2 operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator A on ℋ . Some applications of our inequalities are also given. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the problem of approximation of a function by positive linear operators in weighted spaces. Our main tool is the Pp-statistical convergence recently defined by [M. Ünver and C. Orhan, Numer. Funct. Anal. Optim., 40, 535–547 (2019)]. It is worth noting that the Pp-statistical convergence and statistical convergence do not imply each other. [ABSTRACT FROM AUTHOR]

For a bounded linear operator A on a functional Hilbert space H(O), with normalized reproducing kernel ..., the Berezin symbol and Berezin number are defined respectively by ... and ber(A):= sup .... A simple comparison of these properties produces the inequality ber ... 1/2 = (see [17]). In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces. [ABSTRACT FROM AUTHOR]

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral point spectrum of locally eventually positive semigroups. [ABSTRACT FROM AUTHOR]

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H , which are bounded with respect to the seminorm induced by a positive operator A on H . Here A is not assumed to be invertible. Mainly, if we denote by ω A (⋅) and ω (⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ω A (T) = ω ( A 1 / 2 T (A 1 / 2 ) † ) , where (A 1 / 2 ) † is the Moore-Penrose inverse of A 1 / 2 . In addition, several new inequalities involving ω A (⋅) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496]. [ABSTRACT FROM AUTHOR]

In this article, we consider the subdiffusion equation with the Hadamard fractional derivative of order ρ ∈ (0, 1) and an arbitrary unbounded positive self-adjoint operator A in a Hilbert space. For such equation we study a backward problem. [ABSTRACT FROM AUTHOR]

This article introduces several new relations among related Hilbert space operators. In particular, we prove some Löewner partial orderings among T, |T|, RT, IT, |T|+|T∗| and many other related forms, as a new discussion in this field; where RT and IT are the real and imaginary parts of the operator T . Our approach will be based on proving the positivity of some new matrix operators, where several new forms for positive matrix operators will be presented as a key tool in obtaining the other ordering results. As an application, we present some results treating numerical radius inequalities in a way that extends some known results in this direction, in addition to some results about the singular values. [ABSTRACT FROM AUTHOR]

Given a positive operator A on some Hilbert space, and a nonnegative decreasing summable function μ , we consider the abstract equation with memory u ¨ (t) + A u (t) - ∫ 0 t μ (s) A u (t - s) d s = 0 modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy E (t) = (1 - ∫ 0 t μ (s) d s) ‖ u (t) ‖ 1 2 + ‖ u ˙ (t) ‖ 2 + ∫ 0 t μ (s) ‖ u (t) - u (t - s) ‖ 1 2 d s of any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel μ . This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel μ and of the initial data. [ABSTRACT FROM AUTHOR]

The present work considers two important convergence techniques, namely, deferred type statistical convergence and P$$ P $$‐summability method in respect of linear positive operators. With regard to these techniques, following closely the ideas developed in the articles (Appl. Math. Lett. 18, 2005, 1339‐1344, and Sau Paulo J. Math. Sci. 13, 2019, 696‐707), we state and prove two general non‐trivial Korovkin‐type approximation results for positive linear operators. Further, we define an operator based on multivariate q$$ q $$‐Lagrange–Hermite polynomials and exhibit the applicability of the above theorems to these operators. [ABSTRACT FROM AUTHOR]

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions, together with strictly positive linear maps of matrices. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions, by allowing positive operator monotone decreasing functions instead of power maps.Our proof is based on variational formula for trace functionals based on the Legendre transform, and a strengthened convexity of positive operator monotone decreasing functions in a previous work of Kirihata and the second named author. We also provide the generalization to the framework of unital tracial C -algebras based on Petz's work. [ABSTRACT FROM AUTHOR]

In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples of A-bounded linear operators acting on a complex Hilbert space H , which leads to the special case of the well-known A-numerical radius for d = 1 . Here, A is a positive operator on H. Some inequalities related to the Euclidean operator A-seminorm of d-tuples of A-bounded operators are proved. In addition, under appropriate conditions, several reverse bounds for the A-numerical radius in single and multivariable settings are also stated. [ABSTRACT FROM AUTHOR]

In a series of papers, most of them authored or co-authored by H. Gonska, several authors investigated problems concerning the composition and decomposition of positive linear operators defined on spaces of functions. For example, given two operators with known properties, A and B, we can find the properties of the composed operator A ∘ B , such as the eigenstructure, the inverse, the Voronovskaja formula, and the second-order central moments. One motivation for studying composed operators is the possibility to obtain better rates of approximation and better Voronovskaja formulas. Our paper will address such problems involving compositions of some classical positive linear operators. We present general results as well as numerical experiments. [ABSTRACT FROM AUTHOR]

A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases. [ABSTRACT FROM AUTHOR]

Grover's unstructured quantum search algorithm gives a quadratic speedup over classical linear search, provided that multiple accesses to the oracle are possible. In this paper, we show that in dealing with NP-complete decision-version problems, the quantum computing paradigm still outperforms classical computation, even if only one invocation of the oracle is allowed. The superiority of the quantum approach under such a restrictive condition can often be maintained even if a simple measurement strategy is chosen. A quantum decider utilizing only Hadamard gates and measurements in the computational basis has a better chance of discriminating between a problem with solution(s) and one without, when compared to the best separation achieved by a classical decider. In addition, the simple quantum measurement strategy is remarkably close to the optimal discriminating measurement, which itself is considerably more complex to devise and implement. If we further require the decider to be unambiguous (any definitive answer must be error-free), then a general positive operator-valued measurement can be devised to classify a problem as unsolvable, with some probability, after one consultation of the oracle. Such a feature remains impossible for a classical decider, even after any less than exhaustive oracle invocations. The inherent probabilistic nature of quantum mechanics seems to be at the heart of the advantage quantum computing exhibits over classical computation, in the context of deciding NP-complete problems based on a single oracle query. [ABSTRACT FROM AUTHOR]

We construct three-parameter families of solutions of hyperbolic differential-difference equations in a half-space with a general shift operator in the free term (or in a nonlocal operator potential). It is proved that the solutions obtained are classical if the real part of the symbol of the corresponding differential-difference operators is positive. Classes of equations for which the indicated condition is satisfied are given. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a modification of Szfisz operators involving generalized Brenke type polynomials, First, we investigate Korovkin's type approximation theorem, Next, we established an approximation result for the functions in Lipschitz type space, Furthermore, we discuss the rate of convergence by means of the Ditzian-Totik modulus of smoothness, Lastly, we established an approximation result for the functions having derivatives of bounded variation. [ABSTRACT FROM AUTHOR]

Given scalars an(≠ 0) and bn, n≤0, the tridiagonal kernel or band kernel with bandwidth 1 is the positive definite kernel k on the open unit disc D defined by ... This defines a reproducing kernel Hilbert space Hk (known as tridiagonal space) of analytic functions on D with ... as an orthonormal basis. We consider shift operators Mz on Hk and prove that Mz is left-invertible if and only if {/an/an+1/}n≤0 is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel k, as above, is preserved under Shimorin models if and only if b0 = 0 or that Mz is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fail to yield tridi- agonal Aluthge transforms of shifts defined on tridiagonal spaces. [ABSTRACT FROM AUTHOR]

In this paper, we modified Bernstein-Chlodowsky operators via weaker condition than the classical Bernstein-Chlodowsky operators' condition. We get more powerful results than classical ones. We obtain aproximation properties for these positive linear operators and their generalizations in this work. The rate of convergence of these operators is calculated by means of the modulus of continuity and Lipschitz class of the functions of f of two variables. Finally, we give some concluding remarks with q-calculus. [ABSTRACT FROM AUTHOR]

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form. [ABSTRACT FROM AUTHOR]

A certain discrete probability distribution was considered in ["A discrete probability distribution and some applications", Mediterr. J. Math., 2023]. Its basic properties were investigated and some applications were presented. We now embed this distribution into a family of discrete distributions depending on two parameters and investigate the properties of the new distributions. [ABSTRACT FROM AUTHOR]

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C 2 -smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of locally dominated orthogonally additive operators on lattice-normed spaces (LNS). In the first part of the paper, sufficient conditions for the existence of a local exact majorant of a locally dominated operator and formulas for its calculation are given. The second part shows that the -compactness of a dominated orthogonally additive operator acting from a decomposable lattice-normed space to a Banach space with mixed norm implies the -compactness of its exact majorant. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new upper bounds of ω (T) , which denotes the numerical radius of a bounded operator T on a Hilbert space (H , 〈 · , · 〉) . We show the Aczél inequality in terms of the operator | T | . Next, we give certain inequalities about the A-numerical radius ω A (T) and the A-operator seminorm ∥ T ∥ A of an operator T. We also present several results related to the A -numerical radius of 2 × 2 block matrices of semi-Hilbert space operators, by using symmetric 2 × 2 block matrices. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a general class of operators enriched with some properties in order to act on $ C^{\ast }(\mathbb{R} _{0}^{+}) $. We establish uniform convergence of the operators for every function in $ C^{\ast }(\mathbb{R} _{0}^{+}) $ on $ \mathbb{R} _{0}^{+} $. Then, a quantitative result is proved. A quantitative Voronovskaya-type estimate is obtained. Finally, some applications are provided concerning particular kernel functions. [ABSTRACT FROM AUTHOR]

Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended. [ABSTRACT FROM AUTHOR]

Let A ≥ mA > 0 be a closed positive definite symmetric operator in a Hilbert space ℌ, let and be its Friedrichs and Krein extensions, and let 픖∞ be the ideal of compact operators in ℌ. The following problem has been posed by M.S. Birman: Is the implication A–1 ∈ 픖∞ ⇒ ()–1 ∈ 픖∞(ℌ) holds true or not? It turns out that under condition A–1 ∈ 픖∞ the spectrum of Friedrichs extension might be of arbitrary nature. This gives a complete negative solution to the Birman problem. Let be the reduced Krein extension. It is shown that certain spectral properties of the operators ( + )–1 and P1(I + A)–1 are close. For instance, these operators belong to a symmetrically normed ideal 픖, say are compact, only simultaneously. Moreover, it turns out that under a certain additional condition the eigenvalues of these operators have the same asymptotic. Besides we complete certain investigations by Birman and Grubb regarding the equivalence of semiboubdedness property of selfadjoint extensions of A and the corresponding boundary operators. [ABSTRACT FROM AUTHOR]

We give several further norm inequalities for sums of bounded linear operators on a complex Hilbert space. These inequalities refine and generalize earlier inequalities. Among other applications, it is shown that for A ∈ B (H) and r ∈ [ 0 , 1 ] , w (A) ≤ 1 2 ‖ A ‖ + 1 2 ‖ | A | r | A ⁎ | r ‖ 1 2 ‖ | A | 1 − r | A ⁎ | 1 − r ‖ 1 2 , which refines a result of Kittaneh (2003) [5]. Here, w (⋅) and ‖ ⋅ ‖ denote the numerical radius and the usual operator norm, respectively. [ABSTRACT FROM AUTHOR]

We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descriptions of more tractable sets of elements, where the corresponding averaging can be done arbitrarily close to the centre. This is achieved through several "spectral conditions", involving numerical ranges and tracial states. [ABSTRACT FROM AUTHOR]

We classify rational simple self-dual vertex operator algebras V of CFT and of finite type with positive central charges, where d V = dim ⁡ V 1 is either 8 or 16, under the condition that the space spanned by characters of V -modules is equal to the solution space of some monic modular linear differential equation of the third order. It is shown that central charges are c = 8 for d V = 8 , and c = 4 , 16 for dim ⁡ d V = 16 , respectively. In fact, under the condition that V 1 equipped with the 0th product of V are abelian Lie algebras for d V = 8 , c = 8 and d V = 16 , c = 16 , the vertex operator algebras V are isomorphic to V 2 E 8 and V Λ 16 , respectively. For d V = 16 and c = 4 , V is isomorphic to the lattice vertex operator algebra V A 2 ⊕ A 2 . [ABSTRACT FROM AUTHOR]

It is shown, among other inequalities, that if A and B are Hilbert space operators which belong to the Schatten p -class, then, for p ≥ 1 and 0 < v < 1 , we have ‖ A + B ‖ p ≤ max ⁡ (2 , 2 p − 1) w p , v p ([ 0 A B ⁎ 0 ]) − | 1 − 2 v | p ‖ A − B ‖ p p p and min ⁡ (2 , 2 p − 1) w p , v p ([ 0 A B ⁎ 0 ]) − | 1 − 2 v | p | ‖ A ‖ p − ‖ B ‖ p | p p ≤ ‖ A ‖ p + ‖ B ‖ p , where ‖ ⋅ ‖ p denotes the Schatten p -norm and w p , v is the generalized weighted numerical radius induced by the Schatten p -norm. [ABSTRACT FROM AUTHOR]

In this paper we shall introduce a new type of Bernstein Durrmeyer operators which are not positive on the entire interval [0, 1]. For these operators we will study the uniform convergence on all continuous functions on [0, 1] as well as a result given in terms of modulus of continuity ω(f, δ). A Voronovskaja type theorem will be proved as well. [ABSTRACT FROM AUTHOR]

Since the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor's formula which is a very particular case of Lagrange-Hermite interpolation formula, in the recent paper Gal [3], I have obtained semi-discrete quantitative Voronovskaya-type theorems based on other Lagrange-Hermite interpolation formulas, like Lagrange interpolation on two and three simple knots and Hermite interpolation on two knots, one simple and the other one double. In the present paper we obtain a semi-discrete quantitative Voronovskaya-type theorem based on Lagrange interpolation on arbitrary p + 1 simple distinct knots. [ABSTRACT FROM AUTHOR]

This research concerns some theoretical and numerical aspects of hybrid positive linear operators for approximating continuous functions on [ 0 , 1 ] that have unbounded derivatives at the initial point. These operators are defined by using Modified Bernstein–Kantorovich operators K n , α , where n is positive integer, α > 0 is a fixed constant and reduces to the classical Bernstein–Kantorivich operators when α = 1. To show the importance and the applicability of the given hybrid operators we develop an algorithm which implements them for solving the second kind linear Volterra integral equations with weakly singular kernels. Furthermore, applications are also performed on first kind integral equations, by utilizing regularization. Eventually, it is shown that the numerical realization of the given algorithm is easy and computationally efficient and gives accurate approximations to nonsmooth solutions. [ABSTRACT FROM AUTHOR]

In this work, we obtain some new integral inequalities of the Hermite–Hadamard–Fejér type for operator ω 1 , ω 2 -preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator ω 1 , ω 2 -preinvex functions of the positive self-adjoint operator in the complex Hilbert spaces. We give the special cases to our results; thus, the established results are generalizations of earlier work. In the last section, we give applications for synchronous (asynchronous) functions. [ABSTRACT FROM AUTHOR]

In this article, by considering T = (T1,..., Tn), an n-tuple of commuting contractions on a Hilbert space H, we study T-Toeplitz operators which consists of bounded operators X on H such that Tt*XTt = X for all i = 1..., n. We show that any positive T-Toeplitz operator can be factorized in terms of an isometric pseudo-extension of T. A similar factor-ization result in terms of a BCL type of co-isometric pseudo-extension is also obtained for positive pure lower T-Toeplitz operators. However, a certain dif-ference has been observed between the case n = 2 and n > 2. In a more general context, by considering n-tuples of commuting contractions S and T, we also study (S, T)-Toeplitz operators. [ABSTRACT FROM AUTHOR]

In 1991, Peres and Wootters took an example of detecting the two-copy trine ensemble to demonstrate the nonlocal processing of quantum information (Phys Rev Lett 66:1119, 1991). We in this paper present a complete solution to the classic problem of detecting the N-copy trine ensemble with the minimum-error discrimination strategy. We construct the most general operators of a positive operator valued measure, from which the optimal detection of the N-copy trine ensemble can be derived. The states of the measurement operators are all entangled states, and the correct probability of detecting N-copy trine states tends to unit as the number N of the copies goes to infinite. Our results demonstrate "nonlocality without entanglement" in N-qubit pure states. [ABSTRACT FROM AUTHOR]

The authors analyze approximate characteristics of generalized Poisson operators on Zygmund classes Zα, with the aim of their further application in the theory of optimal decisions. Nowadays, Zygmund classes Zα are increasingly used in optimization methods, which makes the problem important. An estimate of the upper bound of the deviation of functions of the Zygmund class Zα from their generalized Poisson operators in the uniform metric is obtained. Generalized Poisson operators as solutions of the corresponding elliptic partial differential equations are positive linear operators and, therefore, they implement asymptotic approximation of functions of the class Zα in the best way. That is, we get a specific implementation of the optimization problems using the methods of approximation theory. [ABSTRACT FROM AUTHOR]

First, this paper provides characterizing the existence and uniqueness of the linear operator solution T for large classes of full Markov moment problems on closed subsets F of R n. One uses approximation by special nonnegative polynomials. The case when F is compact is studied. Then the cases when F = R n and F = R + n are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of L ν 1 (R n) , and respectively of L ν 1 (R + n) , for a large class of measures ν. One solves the important difficulty created by the fact that on R n ,   n ≥ 2 , there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples. [ABSTRACT FROM AUTHOR]

We consider operator V on the reproducing kernel Hilbert space H = H (Ω) over some set Ω with the reproducing kernel KH,λ (z) = K (z, λ) and define A-Davis-Wielandt-Berezin radius ηA (V ) by the formula ηA (V ) := sup { √ | ⟨V kH,λ, kH,λ⟩ A |² + ∥ V kH,λ ∥A4 : λ ∈ Ω } and ~V is the Berezin symbol of V where any positive operator A-induces a semi-inner product on H is defined by ⟨x, y⟩A = ⟨Ax, y⟩ for x, y ∈ H. We study equality of the lower bounds for A-Davis-Wielandt-Berezin radius mentioned above. We establish some lower and upper bounds for the A-Davis-WielandtBerezin radius of reproducing kernel Hilbert space operators. In addition, we get an upper bound for the A-Davis-Wielandt-Berezin radius of sum of two bounded linear operators. [ABSTRACT FROM AUTHOR]

We propose an approach to a posteriori improvement of approximations in projection methods for solving linear equations with selfadjoint positive definite operators. In particular, we consider the finite element method for two-dimensional boundary value problems. [ABSTRACT FROM AUTHOR]

This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of block matrices, and especially their positivity properties. This is done by focussing on one major problem: characterisation of operators whose numerical radius is bounded by one. So, the article could serve as an introduction to that topic as well. [ABSTRACT FROM AUTHOR]

Kubo and Ando [10] defined a family of operations on positive operators that are now referred to as Kubo-Ando means. These operations are monotone with respect to the usual order and they satisfy the so called transformer inequality, a property which relates the operations to the usual order and congruence transformations. In this paper, we derive a spectral order analogue of the transformer inequality, and we prove that, essentially, the only operations on positive operators which are monotone with respect to the spectral order and satisfy this new transformer inequality are the join and the meet operations with respect to the spectral order. [ABSTRACT FROM AUTHOR]

In this paper we propose some generalizations of real power form for Young-type inequalities which are much better than the recent results developed by many researchers. Our approaching method, together with idea on linear interpolation of Choi et al. (2017) [3] , allows us to build very strict improvements of real power form for Young-type inequalities. As applications, we give the operator versions and inequalities involving unitarily invariant norms and determinants of matrices. [ABSTRACT FROM AUTHOR]